Finite Group Databasebeta

← #26147549

Subgroups — #26147549

Order 12032.

2134 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689025C211yesmaximal, characteristic
2C2×#15689024C211yesmaximal, characteristic
3C2×#15689024C211yesmaximal, characteristic
4C2×#15689022C211yesmaximal, characteristic, complemented
5C2×#15689015C214yesmaximal, complemented
6C2×C42×Dic47C211yesmaximal, characteristic, complemented
7#20241624C214yesmaximal, complemented
8#20241617C211yesmaximal, characteristic, complemented
9C47×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D188C2211yescharacteristic
11C22×C4⋊Dic47C2212yes—
12C22×C4⋊Dic47C2211yescharacteristic
13C22×(C2×D47)⋊C4C2212yes—
14C22×(C2×D47)⋊C4C2214yes—
15C22×(C2×D47)⋊C4C2212yes—
16C22×(C2×D47)⋊C4C2214yes—
17C22×C4×Dic47C2212yes—
18C22×C4×Dic47C2211yescharacteristic
19C22×C4×Dic47C2211yescharacteristic
20C22×C4×Dic47C2212yes—
21#15689025—24no—
22#15689024—24no—
23#15689024C418yescomplemented
24#15689022—24no—
25#15689015—216no—
26C2×C47⋊#95C2214yes—
27C2×C47⋊#95C2212yes—
28C42×Dic47—24no—
29C22×C4⋊C4×C47C2211yescharacteristic
30C22×C4⋊C4×C47C2211yescharacteristic
31C22×C42×C47C2211yescharacteristic
32C2×#95×C47C2214yes—
33C24×D47C2312yes—
34C22×D188—22no—
35C22×D188—22no—
36C22×D188—22no—
37C22×D188—22no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188C2×C414yes—
41C23×Dic47C2312yes—
42C23×Dic47C2312yes—
43C23×Dic47C2311yescharacteristic
44C23×Dic47C2311yescharacteristic
45C2×C4⋊Dic47—22no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—22no—
48C2×C4⋊Dic47—28no—
49C2×C4⋊Dic47—24no—
50C2×(C2×D47)⋊C4—28no—
51C2×(C2×D47)⋊C4—28no—
52C2×(C2×D47)⋊C4—28no—
53C2×(C2×D47)⋊C4—28no—
54C2×(C2×D47)⋊C4—28no—
55C2×(C2×D47)⋊C4—28no—
56C2×(C2×D47)⋊C4—24no—
57C2×(C2×D47)⋊C4—24no—
58C2×(C2×D47)⋊C4—28no—
59C2×(C2×D47)⋊C4C2×C4116yes—
60C2×C4×Dic47—24no—
61C2×C4×Dic47—22no—
62C2×C4×Dic47—22no—
63C2×C4×Dic47—24no—
64C2×C4×Dic47—22no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47—24no—
67C2×C4×Dic47—24no—
68C2×C4×Dic47—24no—
69C2×C4×Dic47C2×C414yes—
70C2×C4×Dic47D414yes—
71C2×C4×Dic47D418yescomplemented
72C47⋊#95—28no—
73C47⋊#95—24no—
74C23×C4×C47C2312yes—
75C23×C4×C47C2312yes—
76C23×C4×C47C2312yes—
77C23×C4×C47C2311yescharacteristic
78C2×C4⋊C4×C47—22no—
79C2×C4⋊C4×C47—22no—
80C2×C4⋊C4×C47—22no—
81C2×C4⋊C4×C47—22no—
82C2×C4⋊C4×C47—22no—
83C2×C4⋊C4×C47C2×C414yes—
84C2×C42×C47—22no—
85C2×C42×C47—22no—
86C2×C42×C47D414yes—
87#95×C47—28no—
88C24×C47C2411yescharacteristic
89C23×D47—24no—
90C23×D47—22no—
91C23×D47—22no—
92C23×D47—24no—
93C23×D47—24no—
94C23×D47—24no—
95C23×D47—22no—
96C23×D47—22no—
97C23×D47—22no—
98C23×D47—22no—
99C23×D47C22×C414yes—
100C22×C4×C47—22no—
101C22×C4×C47—22no—
102C22×C4×C47—22no—
103C22×C4×C47—22no—
104C22×C4×C47—22no—
105C22×C4×C47—24no—
106C22×C4×C47—22no—
107C22×C4×C47—24no—
108C22×C4×C47—22no—
109C22×C4×C47—24no—
110C22×C4×C47—22no—
111C22×C4×C47—22no—
112C22×C4×C47—24no—
113C22×C4×C47C22×C414yes—
114C22×C4×C47C22×C412yes—
115C22×C4×C47C2×D414yes—
116C22×C4×C47C2×D414yes—
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47#4114yes—
120C22×C4×C47#4112yes—
121C22×C4×C47#4112yes—
122C22×C4×C47#4112yes—
123C22×C4×C47#4112yes—
124C22×Dic47—22no—
125C22×Dic47—21no—
126C22×Dic47—21no—
127C22×Dic47—22no—
128C22×Dic47—22no—
129C22×Dic47—22no—
130C22×Dic47—21no—
131C22×Dic47—21no—
132C22×Dic47—22no—
133C22×Dic47—22no—
134C22×Dic47—24no—
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—24no—
138C22×Dic47—22no—
139C22×Dic47C22×C414yes—
140C22×Dic47C2×D414yes—
141C22×Dic47C2×D412yes—
142C22×Dic47C2×D412yes—
143C22×Dic47C2×D414yes—
144C22×Dic47#4114yes—
145C22×Dic47#4114yes—
146C2×D188—28no—
147C2×D188—28no—
148C2×D188—28no—
149C2×D188—44no—
150C2×D188—44no—
151C2×D188—44no—
152C2×D188—44no—
153C4⋊C4×C47—28no—
154C4⋊C4×C47—44no—
155C42×C47—28no—
156(C2×D47)⋊C4—232no—
157(C2×D47)⋊C4—416no—
158(C2×D47)⋊C4—48no—
159(C2×D47)⋊C4—48no—
160C4⋊Dic47—44no—
161C4⋊Dic47—48no—
162C4×Dic47—216no—
163C4×Dic47—28no—
164C4×Dic47—28no—
165C4×Dic47—216no—
166C23×C47C23×C411yescharacteristic
167C23×C47C22×D411yescharacteristic
168C23×C47C22×D412yes—
169C23×C47C22×D411yescharacteristic
170C23×C47C2×#4112yes—
171C23×C47C2×#4112yes—
172C23×C47C2×#4111yescharacteristic
173C23×C47C2×#4111yescharacteristic
174C23×C47C2×#4111yescharacteristic
175C23×C47C2×#4111yescharacteristic
176C23×C47C2×#4112yes—
177C22×D47—24no—
178C22×D47—28no—
179C22×D47—24no—
180C22×D47—24no—
181C22×D47—28no—
182C22×D47—44no—
183C22×D47—42no—
184C22×D47—42no—
185C22×D47—44no—
186C22×D47—42no—
187C22×D47—42no—
188C22×D47—42no—
189C22×D47—44no—
190C22×D47—44no—
191C22×D47—44no—
192C22×D47—44no—
193C22×D47—42no—
194C22×D47—44no—
195C22×D47—44no—
196C22×D47—44no—
197C22×D47—42no—
198C22×D47—44no—
199C22×D47—42no—
200C2×C4×C47—24no—
201C2×C4×C47—24no—
202C2×C4×C47—24no—
203C2×C4×C47—24no—
204C2×C4×C47—24no—
205C2×C4×C47—24no—
206C2×C4×C47—28no—
207C2×C4×C47—24no—
208C2×C4×C47—42no—
209C2×C4×C47—44no—
210C2×C4×C47—42no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—44no—
215C2×C4×C47—44no—
216C2×C4×C47#11714yes—
217C2×C4×C47C4×D414yes—
218C2×C4×C47C4×D414yes—
219C2×C4×C47#12118yes—
220C2×C4×C47#12114yes—
221C2×C4×C47#12314yes—
222C2×C4×C47#12414yes—
223C2×C4×C47#12614yes—
224C2×Dic47—24no—
225C2×Dic47—24no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—24no—
229C2×Dic47—24no—
230C2×Dic47—24no—
231C2×Dic47—24no—
232C2×Dic47—24no—
233C2×Dic47—24no—
234C2×Dic47—24no—
235C2×Dic47—22no—
236C2×Dic47—24no—
237C2×Dic47—22no—
238C2×Dic47—42no—
239C2×Dic47—44no—
240C2×Dic47—42no—
241C2×Dic47—44no—
242C2×Dic47—42no—
243C2×Dic47C4×D418yes—
244C2×Dic47#12414yes—
245C2×Dic47#12714yes—
246D188—416no—
247#14700—471nomaximal, Sylow
248C22×C47C2×#11711yescharacteristic
249C22×C47C2×C4×D412yes—
250C22×C47C2×C4×D411yescharacteristic
251C22×C47C2×C4×D411yescharacteristic
252C22×C47C4×#4112yes—
253C22×C47C2×#12111yescharacteristic
254C22×C47C2×#12112yes—
255C22×C47C2×#12311yescharacteristic
256C22×C47#52512yes—
257C22×C47C2×#12411yescharacteristic
258C22×C47C2×#12411yescharacteristic
259C22×C47C2×#12611yescharacteristic
260C22×C47#52912yes—
261C22×C47#52912yes—
262C22×C47#52912yes—
263C22×C47#52912yes—
264C22×C47C2×#12711yescharacteristic
265C22×C47#53212yes—
266C22×C47#53212yes—
267C22×C47#53212yes—
268C22×C47#53211yescharacteristic
269C22×C47#53212yes—
270C22×C47#53211yescharacteristic
271C2×D47—44no—
272C2×D47—44no—
273C2×D47—48no—
274C2×D47—48no—
275C2×D47—44no—
276C2×D47—82no—
277C2×D47—84no—
278C2×D47—82no—
279C2×D47—82no—
280C2×D47—84no—
281C2×D47—82no—
282C4×C47—28no—
283C4×C47—44no—
284C4×C47—82no—
285C4×C47#38818yescomplemented
286Dic47—24no—
287Dic47—24no—
288Dic47—28no—
289Dic47—82no—
290C2×#390—471no—
291C2×#390—471no—
292C2×#390—471no—
293C2×#389—471no—
294C2×#388—474no—
295C2×C43—471no—
296#1437—474no—
297C4⋊#95—471no—
298C4⋊#95D4711yescharacteristic, complemented
299C2×C47C2×#38811yescharacteristic
300C2×C47#229312yes—
301C2×C47C4×#12112yes—
302C2×C47C4×#12411yescharacteristic
303C2×C47C4×#12711yescharacteristic
304C2×C47#285011yescharacteristic
305C2×C47#285412yes—
306C2×C47#285611yescharacteristic
307C2×C47#285811yescharacteristic
308C2×C47#286012yes—
309C2×C47#286111yescharacteristic
310D47—84no—
311C23×D4—471no—
312C22×C4⋊C4—471no—
313C22×C4⋊C4—472no—
314C22×C4⋊C4C2×D4711yescharacteristic
315C22×C4⋊C4C2×D4711yescharacteristic
316C22×C22⋊C4—474no—
317C22×C22⋊C4—472no—
318C22×C22⋊C4—474no—
319C22×C22⋊C4—472no—
320C22×C42—471no—
321C22×C42—472no—
322C22×C42—471no—
323C22×C42—472no—
324C22×C42C2×D4711yescharacteristic
325#390—478no—
326#390—944no—
327#390—944no—
328#389—944no—
329#388—9416no—
330C2×#95—474no—
331C2×#95—472no—
332C2×#95C2×D4714yes—
333C43—944no—
334C47#1470011yescharacteristic, complemented, Sylow
335C25—472no—
336C22×D4—474no—
337C22×D4—942no—
338C22×D4—942no—
339C22×D4—942no—
340C22×D4—942no—
341C22×D4—942no—
342C22×D4—942no—
343C23×C4—471no—
344C23×C4—472no—
345C23×C4—471no—
346C23×C4—472no—
347C23×C4C22×D4712yes—
348C23×C4C22×D4712yes—
349C23×C4C22×D4712yes—
350C23×C4C22×D4711yescharacteristic
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—948no—
357C2×C4⋊C4—944no—
358C2×C4⋊C4—942no—
359C2×C4⋊C4—942no—
360C2×C4⋊C4—942no—
361C2×C4⋊C4C4×D4714yes—
362C2×C22⋊C4—4716no—
363C2×C22⋊C4—948no—
364C2×C22⋊C4—948no—
365C2×C22⋊C4—948no—
366C2×C22⋊C4—948no—
367C2×C22⋊C4—944no—
368C2×C22⋊C4—944no—
369C2×C22⋊C4—948no—
370C2×C22⋊C4—948no—
371C2×C22⋊C4—948no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—474no—
375C2×C42—474no—
376C2×C42—478no—
377C2×C42—944no—
378C2×C42—944no—
379C2×C42—944no—
380C2×C42—942no—
381C2×C42—944no—
382C2×C42—942no—
383C2×C42—942no—
384C2×C42—944no—
385C2×C42—942no—
386C2×C42D18814yes—
387#95—28no—
388#95—948no—
389#95—944no—
390C24—474no—
391C24—944no—
392C24—942no—
393C24—942no—
394C24—944no—
395C24—944no—
396C24—942no—
397C24—942no—
398C24—942no—
399C24—944no—
400C24—942no—
401C24C23×D4711yescharacteristic
402C2×D4—948no—
403C2×D4—948no—
404C2×D4—948no—
405C2×D4—1884no—
406C2×D4—1884no—
407C2×D4—1884no—
408C2×D4—1884no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—474no—
423C22×C4—474no—
424C22×C4—472no—
425C22×C4—474no—
426C22×C4—472no—
427C22×C4—474no—
428C22×C4—474no—
429C22×C4—942no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—942no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—942no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—942no—
439C22×C4—944no—
440C22×C4—942no—
441C22×C4—942no—
442C22×C4—944no—
443C22×C4—942no—
444C22×C4C2×C4×D4714yes—
445C22×C4C2×C4×D4712yes—
446C22×C4C2×D18814yes—
447C22×C4C2×D18812yes—
448C22×C4#1062535012yes—
449C22×C4D4×D4714yes—
450C22×C4D4×D4712yes—
451C22×C4#1062535212yes—
452C22×C4#1062535212yes—
453C22×C4#1062535414yes—
454C22×C4#1062535412yes—
455C4⋊C4—28no—
456C4⋊C4—44no—
457C4⋊C4—1888no—
458C4⋊C4—1884no—
459C22⋊C4—9432no—
460C22⋊C4—1888no—
461C22⋊C4—1888no—
462C22⋊C4—18816no—
463C42—28no—
464C42—9416no—
465C42—948no—
466C42—9416no—
467C42—948no—
468C23—944no—
469C23—948no—
470C23—944no—
471C23—944no—
472C23—948no—
473C23—1884no—
474C23—1882no—
475C23—1882no—
476C23—1884no—
477C23—1882no—
478C23—1882no—
479C23—1882no—
480C23—1884no—
481C23—1884no—
482C23—1884no—
483C23—1884no—
484C23—1882no—
485C23—1884no—
486C23—1884no—
487C23—1884no—
488C23—1882no—
489C23—1884no—
490C23—1882no—
491C23C22×C4×D4711yescharacteristic
492C23C2×D4×D4712yes—
493C23C2×D4×D4711yescharacteristic
494C23C22×D18811yescharacteristic
495C23#41×D4712yes—
496C23#41×D4712yes—
497C23C2×#1062535211yescharacteristic
498C23C2×#1062535211yescharacteristic
499C23C2×#1062535011yescharacteristic
500C23C2×#1062535411yescharacteristic
501C23C2×#1062535412yes—
502D4—18816no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—478no—
520C2×C4—474no—
521C2×C4—474no—
522C2×C4—944no—
523C2×C4—944no—
524C2×C4—942no—
525C2×C4—942no—
526C2×C4—944no—
527C2×C4—944no—
528C2×C4—944no—
529C2×C4—944no—
530C2×C4—944no—
531C2×C4—944no—
532C2×C4—944no—
533C2×C4—942no—
534C2×C4—944no—
535C2×C4—942no—
536C2×C4—1884no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4—1884no—
540C2×C4—1882no—
541C2×C4#1309398214yes—
542C2×C4(C22⋊D47)⋊C414yes—
543C2×C4C4×D18814yes—
544C2×C4#1309399918yes—
545C2×C4#1309399914yes—
546C2×C4#1309400814yes—
547C2×C4#1309401014yes—
548C2×C4#1309401914yes—
549C22—3762no—
550C22—3764no—
551C22—3762no—
552C22—3762no—
553C22—3764no—
554C22—3762no—
555C22—1884no—
556C22—1888no—
557C22—1888no—
558C22—1884no—
559C22—1884no—
560C22C2×#1309398211yescharacteristic
561C22C4×D4×D4712yes—
562C22C2×(C22⋊D47)⋊C411yescharacteristic
563C22C2×C4×D18811yescharacteristic
564C22C4×#1062535412yes—
565C22C2×#1309399911yescharacteristic
566C22C2×#1309399912yes—
567C22C2×#1309400811yescharacteristic
568C22#1568963512yes—
569C22#124×D4711yescharacteristic
570C22C2×#1309401011yescharacteristic
571C22C2×#1309401911yescharacteristic
572C22#1568965812yes—
573C22#1568965912yes—
574C22#1568965912yes—
575C22#1568966012yes—
576C22#127×D4711yescharacteristic
577C22#1568966812yes—
578C22#1568966812yes—
579C22#1568967012yes—
580C22#1568967211yescharacteristic
581C22#1568967412yes—
582C22#1568967411yescharacteristic
583C4—28no—
584C4—44no—
585C4—82no—
586C4—944no—
587C4—944no—
588C4—948no—
589C4—3762no—
590C4#1568901518yescomplemented
591C2—3764no—
592C2C2×#1568901511yescharacteristic
593C2#2024584612yes—
594C2C4×#1309399912yes—
595C2#2024593111yescharacteristic
596C2#2024595211yescharacteristic
597C2#2025169011yescharacteristic
598C2#2025172012yes—
599C2#2025173411yescharacteristic
600C2#2025174211yescharacteristic
601C2#2025175612yes—
602C2#2025176211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C4⋊#95C2
socle, cosocleC24×C47C24
prime radicalC23×D188C22
hypercenterC4⋊#95D47
hypocenterC47#14700
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#15689022C2semidirect
5. C2×#15689015C2semidirect
6. C2×C42×Dic47C2semidirect
7. #20241624C2semidirect
8. #20241617C2semidirect
9. C47×C4⋊#95C2semidirect
23. #15689024C4semidirect
71. C2×C4×Dic47D4semidirect
285. C4×C47#388semidirect
298. C4⋊#95D47semidirect
334. C47#14700semidirect
590. C4#15689015semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156890252
C2×#156890242
C2×#156890222
C2×#156890152
C2×C42×Dic472
#202416242
#202416172
C47×C4⋊#952
#1470047